Question

How many ways can 4 boys and 3 girls be seated in a row such that all girls sit together?

• A. 720
• B. 144
• C. 840
• D. 5040

Hint:

Whenever a question says "sit together," think of them as a single person. Don't forget the second step: once you arrange the group, always unwrap the bundle and arrange the individuals inside it!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem belongs to permutations with constraints. When certain items (in this case, all 3 girls) must always sit together, we can use the "string method" or "block method." We tie the restricted items together into a single composite entity or block, calculate the arrangements of the remaining items along with this block, and then arrange the items inside the block itself.

Step 2: Key Formula or Approach:
1. Number of ways to arrange $n$ distinct objects in a row is $n!$ (factorial). 2. Total arrangements = (Arrangement of the units outside/including the block) $\times$ (Arrangement of individuals inside the block).

Step 3: Detailed Explanation:
Let us group all 3 girls into a single consolidated unit: $(G_1G_2G_3)$. Now, we count this group of girls as 1 single unit, alongside the 4 individual boys ($B_1, B_2, B_3, B_4$). Total number of distinct entities to arrange = $4 \text{ boys} + 1 \text{ group of girls} = 5 \text{ entities}$. The number of ways to arrange these 5 entities in a row is: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \text{ ways} \] Within the girls' unit, the 3 girls can be arranged among themselves in: \[ 3! = 3 \times 2 \times 1 = 6 \text{ ways} \] By the fundamental counting principle, the total number of valid seating arrangements is the product of these two values: \[ \text{Total arrangements} = 5! \times 3! = 120 \times 6 = 720 \]

Step 4: Final Answer:
The number of ways they can be seated is 720.